Interaction of the Beryllium Cation with Molecular Hydrogen and

Article pubs.acs.org/JPCA

Interaction of the Beryllium Cation with Molecular Hydrogen and Deuterium Denis G. Artiukhin,† Jacek Kłos,‡ Evan J. Bieske,§ and Alexei A. Buchachenko*,† †

Department of Chemistry, Moscow State University, Moscow 119991, Russia Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742-2021, United States § School of Chemistry, The University of Melbourne, Parkville, VIC 3010, Australia ‡

ABSTRACT: The structural and spectroscopic properties of the Be+−H2 and Be+−D2 electrostatic complexes are investigated theoretically. A three-dimensional ground-state potential energy surface is generated ab initio at the CCSD(T) level and used for calculating the lower rovibrational energy levels variationally. The minimum of the potential energy surface corresponds to a well depth of 3168 cm−1, an intermolecular separation of 1.776 Å, with the bond of the H2 subunit being 0.027 Å longer than for the free molecule. Taking vibrational zero point energy into account, the complexes containing para H2 and ortho D2 are predicted to have dissociation energies of 2678 and 2786 cm−1, respectively. The νHH band of Be+− H2 is predicted to be red-shifted from the free dihydrogen transition by −323 cm−1, whereas the corresponding shift for Be+−D2 is predicted to be −229 cm−1. The dissociation energy of the Be+−D2 complex is calculated to be slightly higher than the energy required to vibrationally excite the D2 subunit, raising the possibility that the onset of dissociation can be observed in the infrared predissociation spectrum at a particular rotational energy level in the νDD manifold.

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INTRODUCTION Understanding the interactions between molecular hydrogen and atomic ions is of interest in many respects. Adsorption of hydrogen on acidic or basic centers of porous materials is a key step of current industrial chemical processes including catalytic hydrogenation/dehydrogenation and is expected to be important for future hydrogen storage technologies. Two basic parameters, the heat of adsorption and degree of hydrogen bond activation (qualitatively related to its elongation Δr or vibrational frequency shift ΔνHH with respect to the gasphase value), are of prime importance for customizing the materials to practical demands. Organometallic chemistry has provided the means to control these parameters over a wide range.1 For rationally designing new materials, the interactions of the hydrogen molecule with a bare ion in the gas phase provides a useful reference point and benchmark. It is therefore not surprising that significant efforts have been devoted to the experimental characterization complexes consisting of dihydrogen attached to a charged atom, as summarized in a recent review.2 The electrostatic M+−H2 and A−−H2 complexes are formed due to long-range electrostatic and induction forces (charge− quadrupole and charge−induced dipole, to the lowest order). The electrostatic interaction largely dictates the equilibrium structure of the complex − T-shaped for cations and linear for anions.2,3 However, more subtle chemical effects augment the electrostatic and induction interactions and determine large variations of the structural and energetic parameters. These © 2014 American Chemical Society

effects have been discussed for complexes containing transition metal cations (see, e.g., refs 4−6) prompted by their importance for catalysis and the ion−molecular chemistry of d-elements.7,8 The interaction strength, characterized by the well depth De or dissociation energy D0, determines the heat of the complex formation, whereas Δr or ΔνHH are symptomatic of the perturbation to the hydrogen molecule. These quantities are of interest for direct correlations with adsorption properties. Thermochemical methods are convenient for determining binding energies and most data originate from clustering equilibrium measurements.9−11 During the past decade, a series of M+−H2 complexes have been probed using infrared (IR) photodissociation spectroscopy, extending our understanding of interactions between metal cations and molecular hydrogen.2,12 The IR spectra of M+−H2 complexes have been obtained by exciting the nHH = 0 → nHH = 1 transition of the H2 diatomic with detection of the M+ ionic photofragments. The ensuing spectra display rotational substructure reflecting the geometry of the complexes and providing the frequency shift ΔνHH. Generally, spectroscopic studies do not directly reveal the dissociation energy. An exception is the Cr+−D2 complex, for which there is an onset Special Issue: Franco Gianturco Festschrift Received: May 4, 2014 Revised: June 30, 2014 Published: June 30, 2014 6711

dx.doi.org/10.1021/jp504363d | J. Phys. Chem. A 2014, 118, 6711−6720

The Journal of Physical Chemistry A

Article

calculations, and the properties of stationary points. Second, some results for excited-state PESs are briefly mentioned. The interaction energy between the Be cation in its ground 2 S state and the 1Σg+ H2 molecule produces a single groundstate PES of 2A′ symmetry. The geometry of the Be+−H2 complex is conveniently described by Jacobi coordinates (r, R, θ). Here, r denotes the internuclear separation in the H2 molecule, R is the vector from the H2 center-of-mass to the Be+ ion that defines the z-axis of the body-fixed (BF) frame, and θ is the angle between r and R. Ab Initio Calculations. The reference wave function for the Be+−H2 system for subsequent single-reference coupled-cluster calculations was obtained at the restricted Hartree−Fock (RHF) level. The electronic correlation energy was recovered by partially spin-restricted coupled-cluster method including single and double excitations and noniterative triple excitations [CCSD(T)].44 The interaction energy was calculated within a supermolecular approach:45

for photofragmentation at a particular rovibrational level in the nDD = 1 manifold, permitting determination of the dissociation energy more accurately than by thermochemical measurements.13 Owing to developments in the laser cooling of atomic ions, another spectroscopic technique sensitive to the interaction between dihydrogen and metal cations has recently emerged. Collisions between molecular hydrogen and a single, ultracold, laser-trapped M+ ion results in reaction on the excited-state potential energy surface and formation of MH+ product ions (see, e.g., refs 14−16 for Be+ and Mg+ ions). Experimental progress has revitalized interest in theoretical studies of the ion-hydrogen interactions. Although the equilibrium structure and harmonic vibrational frequencies have been calculated ab initio for a large range of complexes,2 generally it is necessary to go beyond the harmonic oscillator and rigid rotor approximations to describe adequately the nonrigid M+−H2 systems. For example, vibrational anharmonicty has been dealt with using the vibrational self-consistent field method.17 Previous experience18−21 including our own,22−29 shows that for the main-group anion and cation electrostatic complexes variational rovibrational energy level calculations based on a three-dimensional (3D) potential energy surface (PES) obtained at a high level of the ab initio theory with correlation treatment reproduces the observed spectral features, and helps assignment and fitting of line positions within conventional spectroscopic models. To account for the H−H stretching vibration, the diabatic separation of the fast diatomic vibration was found to be a good approximation, thereby reducing the vibrational problem to two dimensions, facilitating spectroscopic predictions for high rotational angular momentum J levels and for metastable levels in the excited nHH = 1 manifold. In this paper, we apply this theoretical methodology to make quantitative predictions for the structure, energetics and spectroscopic constants for the Be+−H2 and Be+−D2 electrostatic complexes. Characterizing the interaction between Be+ and H2 is important for understanding the storage of hydrogen in porous materials with oxidized beryllium centers. Despite beryllium being a rare and toxic element, Be-based metal−organic frameworks have been synthesized and tested for hydrogen storage properties,30 and some noninterstitial Be-containing hydrides have been considered for the same purpose.31 Beryllium is also of interest for astrophysics and astrochemistry, and despite its low abundance, the Be+ ion is considered as an important astrophysical tracer. Beryllium’s origin in the Universe is controversial and it is believed to be formed not through Big Bang or stellar nucleosyntheses, but rather through galactic cosmic rays.32−34 The BeH molecule and its cation have been identified in comets and stars.35,36 Neutral BeH2 is a linear symmetric molecule formed by Be insertion into the H2 bond, as established spectroscopically37,38 and theoretically.39,40 In contrast, the Be+−H2 electrostatic complex has not yet been characterized experimentally. However, ab initio calculations presented in refs 41 and 42 and, most recently, by Page et al.,43 show that the T-shaped electrostatic Be+−H2 complex is the most stable structure, with the linear HBeH+ isomer lying more than 10 000 cm−1 higher in energy.

DCBS + V (r ,R ,θ ) = E Be+ − H2(r ,R ,θ ) − E Be (r ,R ,θ )

− E HDCBS (r ,R ,θ ) 2

(1)

where EBe+−H2(r,R,θ) is the total energy of the entire complex DCBS + and EDCBS Be+ (r,R,θ) and EH2 (r,R,θ) are the total energies of Be and H2 monomers, respectively, calculated in the basis set of all constituentsthe dimer centered basis set (DCBS). This procedure accounts for the basis set superposition error using the standard counterpoise (CP) correction scheme46 but excludes the intramolecular potential energy of the H2 moiety at R → ∞, U(r). The correlation-consistent aug-cc-pVQZ basis set of the quadruple-ζ quality47 for hydrogen, extended with additional 3s3p2d2f1g bond functions48 placed halfway between the H2 center of mass and Be cation, was used for the calculations. For the Be cation, the core−valence correlation was included in coupled-cluster calculations by correlating the electrons for the 1s orbital. The specially adapted aug-cc-pCVQZ basis49 was employed for this purpose. The potential energy was calculated on a discrete grid in which the diatomic distance r varies from 1.0 to 2.4 bohr in steps of 0.2 bohr, θ varies from 0° to 90° in steps of 22.5°, and R varies from 2.5 to 50 bohr in 44 points (with varied step from 0.25, 0.5 to 1.0 bohr for the larger distances). The full 3D PES was obtained by summing the interaction PES V(r,R,θ) from eq 1 and an accurate potential energy curve for the H2 monomer U(r).50 All electronic structure calculations were performed with the MOLPRO (ver. 2010.1) program package.51 Analytical Fit. The analytical representation of the PES is similar to the one used in our previous studies of the Mg+−H2 complex.27 In brief, it implements the reproducing kernel Hilbert space (RKHS) method52 for fitting the ab initio points and extrapolating the PES with the radial kernel corresponding to a R−3 long-range dependence; see below. The only adaptation this method requires for the present purpose is the size of the Be+−H2 ab initio grid. The FORTRAN routines for generation of the analytical PES are available from the authors upon request. Characterization of the PES. The two-dimensional contour plot of the PES constructed using the RKHS fit in the X = R cos θ, Y = R sin θ Cartesian coordinates at the equilibrium distance of a free hydrogen molecule re(H2) = 0.741 Å is presented in Figure 1. The features of the PES are

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POTENTIAL ENERGY SURFACES In this section we first describe the electronic structure methods used in the calculations of the ground-state interaction PES, its analytical representation for subsequent bound-state 6712



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Table 1. Stationary Points of the Be+−H2 ab Initio PESa r, Å

method CCSD(T), this work CCSD(T)43 CCSDT43 MRCI+Q, this work MRCI+Q43 CCSD(T), this work MRCI+Q, this work MRCI+Q43

Figure 1. Contour plot of the Be+−H2 PES at r = 0.741 Å in Cartesian (X,Y) coordinates. Solid lines correspond to negative energy contours from −3000 to −300 cm−1, with the increment of 300 cm−1, dotted line corresponds to zero energy (Be+ + H2 dissociation limit), dashed lines correspond to positive contours 5, 10, 15, and 20 cm−1. Arrows indicate the positions of the stationary points, minimum, saddle, and maximum, in order of increasing X.

CCSD(T), this work a

E, cm−1

1.776 1.775 1.776 1.794 1.774

−3168 −3154 −3170 −3020 −3018

2.153 2.174 2.152

−1209 −1185 −1070

5.923

21

Energies are given with respect to the Be+ + H2 dissociation limit.

convergence of our ab initio scheme and justify the perturbative treatment of triple excitations. Interaction with the Be+ ion significantly affects the hydrogen molecule. At the equilibrium configuration of the complex, the H−H bond is elongated by 0.027 Å (4%) compared to the bond in the free H2 molecule. As noted in ref 2 in relation to the results from ref 43, this is a record Δr value for main-group cations. The short equilibrium interfragment separation R and large well depth De = −E are also evidence for the strong interaction between Be+ and molecular hydrogen. Numerical solution of the normal mode vibrational problem for the fitted PES gives a stretching frequency for the H2 subunit of νHH(h) = 4082 cm−1, shifted to the red by 320 cm−1 (7%) with respect to that of the free dihydrogen molecule [hereafter “(h)” indicates harmonic approximation]. Page et al.43 reported an even lower value, 3850 cm−1. This mismatch seems strange in view of the excellent agreement for other parameters. Intermolecular stretching νs(h) and bending νb(h) frequencies for Be+−H2 were calculated as 629 and 804 cm−1, respectively. Thus, the dissociation energy corrected for harmonic zero-point energy amounts to D0(h) = 2452 cm−1. For Be+−D2, the harmonic vibrational frequency shift is ΔνDD(h) = 228 cm−1, whereas νs(h) = 484 cm−1, νb(h) = 571 cm−1, and D0(h) = 2641 cm−1. Excited-State PESs: An Overview. Despite the calculations by Page et al.43 demonstrating very reasonable agreement between the single-reference CCSD(T) and multireference MRCI+Q approaches at the equilibrium configuration, one cannot exclude the failure of the former approach at shorter interfragment distances due to possible interaction with excited electronic states. To investigate this, we performed the test MRCI+Q calculations using the compact aug-cc-pVTZ basis set and active space consisting of 1s, 2s, and 2p orbitals for both Be and H centers. No bond functions and CP correction for basis set superposition error were used. The lowest excited states of the Be+−H2 complex arise from the excitation of the ion to its first excited 2P° term by 31 000 cm−1 (32 200 cm−1 in our calculations).54 In the triatomic complex of Cs symmetry, this excitation gives rise to three adiabatic states, 22A′ and 32A′ with electron excitation to inplane 2pz and 2py orbitals, and 12A″ associated with out-ofplane 2px excitation. One-dimensional sections of the corresponding PESs along the R coordinate (r and θ coordinates were optimized for the ground state at each R value) are shown in Figure 2. Both 22A′ and 12A″ PESs are much more attractive than the ground 12A′ PES. As a result,

typical for M+−H2 complexes and can be understood as resulting from the lowest-order long-range intermolecular forces, namely, the electrostatic charge−quadrupole Vel = ΘqP2(cos θ )/R3

Minimum, θ = 90° 0.768 0.767 0.768 0.767 0.767 Saddle, θ = 0 0.761 0.762 0.761 Maximum, θ = 0 0.741

R, Å

(2)

and the charge−induced dipole induction 1 Vind = − q2[α0 + α2P2(cos θ )]/R4 (3) 2 interactions. Here we use atomic units (au) and denote, for H2, the quadrupole moment as Θ, and the isotropic and anisotropic polarizabilities as α0 = (α∥ + 2α⊥)/3 and α2 = 2(α⊥ − α∥)/3, respectively. The charge of the Be+ ion is q = +1 and Θ for H2 is positive.53 For M+−H2 complexes, the induction interaction is attractive for all orientations but favors a θ = 0 linear configuration (because α∥ > α⊥). The charge−quadrupole interaction is attractive for arccos(1/√3) < θ < π/2 + arccos(1/√3) with a minimum at θ = 90° and is repulsive outside this range. At chemically significant intermolecular separations (R > 1 Å), the charge−quadrupole interaction dominates the induction interaction and favors a T-shaped M+−H2 complex. For the linear configuration (θ = 0), the competition between the charge−quadrupole interaction (which is repulsive) and the induction interaction (which is attractive) leads to appearance of a maximum at long-range and a saddle at shorter distance. Three stationary points are marked in Figure 1 by arrows and are summarized in Table 1 in terms of the r and R distances and energy E, the sum of inter- and intramolecular terms V and U, related to its R → ∞ limit, i.e., free Be+ cation and H2 molecule at equilibrium. The results for the equilibrium structure are in excellent agreement with the calculations by Page et al.43 who used the same basis set for hydrogen, an atomic natural orbital basis set for Be, and no bond functions with calculations using the methods CCSD(T), CCSDT (full account of triples in the single-reference coupled cluster method), and multireference configuration interaction with Davidson correction (MRCI +Q), CP correction included. Good agreement is also found for the saddle point, especially if one takes into account the 150 cm−1 energy difference between CCSD(T) and MRCI+Q results at equilibrium. These comparisons support the 6713



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The states correlating with the Be+(2P°) term are involved in the photoinitiated bond cleavage of molecular hydrogen colliding with Be+ in Coulomb crystals. Roth et al.14 found that this process has a rate constant as high as 10−9 cm3/s, pointing to a direct barrierless reaction pathway. To derive insights into this reaction, we computed 2D sections of the excited PESs as a function of the r and R coordinates at selected values of θ, as shown in Figure 3 for the lowest 22A′ state. The PES is strongly repulsive with respect to formation of the BeH+ molecular ion. The θ = 90° panel in Figure 3 shows the contours associated with the minimum region of the groundstate PES and indicates, in accord with observations, that collisions of excited Be+ with H2 should lead to BeH+ formation with almost unit probability.14 One interesting conclusion of this analysis is that for the Be+−H2 complex it should be possible to initiate the process with photons whose energies are 12 000 cm−1 lower than for the 2S → 2P° transition of the bare Be+ ion. Our calculations also allow the exothermicity of the Be+(2P°) + H2 → BeH+(X1Σ+) + H reaction to be estimated as 2.397 eV. On the other hand, the ground-state reaction should be endothermic by 1.596 eV (cf. 1.57 eV, as calculated in ref 55). One should note that the role of excited states in the Mg+ + H2 reactions was carefully analyzed by Gianturco and coworkers.16 More work is needed to achieve the same degree of detail for the Be+ + H2 reaction. In the remainder of this paper, we concentrate on the ground-state Be+−H2 electrostatic complex.

Figure 2. One-dimensional sections of the lowest Be+ + H2 PESs along the R coordinate. r and θ coordinates were optimized for the ground state at each R value.

avoided crossing of the two lowest 2A′ PESs occurs at R ≈ 1.2 Å, almost at the minimum of the excited curve. For the ground state, however, the crossing point lies high enough on the repulsive branch to justify the validity of the single-reference CCSD(T) method over the whole range of geometries relevant to the ground-state electrostatic complex and its rovibrational bound states.

Figure 3. Contour plots of the 22A′ PES at four values of the θ angle. The contours are labeled by energies (cm−1) with respect to the Be+(2P°) + H2(r=re) limit. Gray contours in the θ = 90° panel characterize the minimum region of the ground-state PES and are labeled with respect to the Be+(2S) + H2(r=re) limit that lies 32 200 cm−1 below. 6714



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For IR photodissociation spectroscopy of M+−H2 complexes, the IR beam excites a nHH = 0 → nHH = 1 transition of the H2 subunit, which leads to vibrational predissociation of the complex and liberation of an M+ fragment, which is detected.2 The levels associated with the nHH = 1 state are metastable and cannot be obtained using the expansion (5) over the squareintegrable radial functions. To estimate the shift of the nHH = 0 → nHH = 1 transitions, we used, as before,22 the so-called vibrational diabatic decoupling approximation.56 This assumes that the fast vibration of the H2 moiety is unaffected by the slow intermolecular dynamics, so its wave function is given by the solution of eq 6 for certain nHH. The intermolecular motion is therefore determined by the effective two-dimensional (2D) Hamiltonian

ROVIBRATIONAL ENERGY LEVELS Variational Method. The RKHS PES was used to calculate the lowest rovibrational energy levels of the Be+−H2 complex using the procedure described in ref 22. The total rovibrational Hamiltonian describing the complex in 3D is written in au as (J − j)2 j2 1 ∂2 1 ∂2 − + + + U (r ) Ĥ = − 2 2 2 2μ ∂R 2m ∂r 2μR 2mr 2 + V (r ,R ,θ )

(4)

where J and j are the rotational angular momenta of the complex and of the H2 diatomic, respectively, whereas μ and m are the corresponding reduced masses. The total PES is given as the sum of the unperturbed intermolecular H2 potential U and intermolecular interaction V in accord with the ab initio approach described above. The total wave function is represented by a linear combination of the products of H2 vibrational χ, radial ϕ, and angular Θ basis functions: JMpp i j

Ψn

=

,n Jpp i j

JMpp i j

∑ ∑ ∑ CwkjΩ χw (r) ϕk(R) Θ jΩ w

k

(J − j)2 1 ∂2 + + BnHH j2 + VnHH(R ,θ ) Ĥ nHH = − 2μ ∂R2 2μR2 + ϵnHH

where BnHH = ⟨χnHH|r−2|χnHH⟩/2m, VnHH = ⟨χnHH|V(r,R,θ)|χnHH⟩r. This approximation is equivalent to retaining the single w = nHH term in eq 5. The variational solution is obtained using the same radial and angular basis sets as described above and provides the same accuracy. Note that the spin−rotation interaction is ignored. As argued and found for the Mg+−H2 complex, the corresponding splitting is small in comparison with the typical line width in the IR photodissociation spectrum.27 Three-Dimensional Calculations. The results of the 3D calculations for the lowest energy levels of complexes consisting of Be+ attached to pH2, oH2, oD2, and pD2 for J = 0 and 1 are presented in Table 2. The levels of the complexes containing pH2 and oD2 appear in the 0++ and 1++ symmetry blocks, whereas those of oH2 and pD2 are in the 1+− and 0+− blocks. All energies are related to the lowest dissociation limit, Be+ + pH2 (nHH = 0,j = 0) or Be+ + oD2 (nDD = 0,j = 0). However, because the Be+−oH2 and Be+−pD2 complexes can only dissociate to give the rotationally excited j = 1 diatomic fragment, their dissociation energy should be corrected by the rotational energy of the product molecule, 118.67 cm−1 for H2 (j = 1) and 59.83 cm−1 for D2 (j = 1). As a result, the dissociation energy for Be+−pH2 is D0 = 2678 cm−1, whereas for Be+−oH2 it is D0 = 2727 cm−1. Dissociation energies of the deuterated complexes are larger due to lower vibrational zero-point energy: D0 = 2786 cm−1 for Be+−oD2 and D0 = 2812 cm−1 for Be+−pD2. This trend, that complexes containing the even-j spin-isomers (pH2, oD2) have lower absolute energy than complexes containing the odd-j spin-isomers (oH2, pD2) but are effectively less stable, is common to all M+−H2 and A−−H2 complexes studied so far.2 Note that using the harmonic approximation for the vibrational frequencies and zero-point energy leads to D0 being underestimated by 5−8%. Table 2 also lists vibrationally averaged distances for each level. The ⟨r⟩ and ⟨R⟩ values were computed as expectation values of r and R with the wave function (5), whereas r ̅ and R̅ were obtained from the expectation values of r−2 and R−2, in a manner akin to spectroscopic structure determination. Computing the same quantities for the free hydrogen molecule, ⟨r⟩ = 0.766 and r ̅ = 0.751 Å, we estimate that the vibrationally averaged H−H bond length increases by 0.012−0.014 Å when Be+ is attached. This is only half the corresponding increase of the H2 equilibrium bond length (Δre = 0.027 Å, Table 1). The

(r·̂ R̂ )

jΩ

(5)

where Ω denotes the absolute value of the projection of J and j in the BF frame and M is the projection of J onto the spacefixed (SF) axis. The χw functions are chosen as the wave functions of the free H2 molecule evaluated numerically: ⎡ 1 ∂2 ⎤ + U (r )⎥χw (r ) = ϵwχw (r ) ⎢− 2 ⎣ 2m ∂r ⎦

(6)

Radial ϕk functions are also computed numerically by solving a similar Schrödinger equation ⎡ 1 ∂2 ⎤ + V (r =re ,R ,θ =90°)⎥ϕk (R ) = ekϕk (R ) ⎢− 2 ⎣ 2μ ∂R ⎦

(9)

(7)

with the one-dimensional radial section of the interaction PES ipj at θ = 90°. The angular basis ΘJMp consists of the symmetrized jΩ 56 rigid rotor functions: ⎡ 2J + 1 ⎤1/2 JMp p Θ jΩ i j(r ·̂ R̂ ) = ⎢ ⎥ ⎣ 8π(1 + δΩ0) ⎦ *JΩ(α ,β ,0) YjΩ(θ ,φ) + p (− 1) J DM *J−Ω(α ,β ,0) Yj −Ω(θ ,φ)] × [DM i (8) J D*MΩ (α,β,0)

where is the Wigner rotational matrix, which depends on the Euler angles connecting BF and SF frames, and YjΩ(θ,φ) are spherical harmonic functions describing 3D rotation of H2 with respect to BF frame. This choice of angular functions accounts for the symmetry by means of the parity quantum numbers.57 The parity index pi = ± is connected to the inversion parity p, p = pi(−1)J, whereas the permutation parity pj defines the parity of j (pj = + for even j and pj = − for odd j). One should note that pj is connected to the symmetry of the nuclear spin wave function of the monomer, namely, pj = + for p-hydrogen (pH2) and o-deuterium (oD2) and pj = − for ohydrogen (oH2) and p-deuterium (pD2). Jp p Variational solution gives the energies En i j and expansion coefficients C in eq 5 for the levels associated with H2 fragment in its ground vibrational state nHH = 0. Convergence tests showed that a basis set consisting of 4 vibrational, 10 radial, and 12 angular functions provides the 6 lowest energy levels of each Jpipj block with 0.02 cm−1 accuracy. 6715



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467 cm−1. The frequencies νs(h) determined by the normalmode analysis are 4−7% higher. For symmetry reasons, only even bending excitations are allowed within the 0++ symmetry block that contains the ground-state level. The 2νb excitation energies amounts to 1218 and 968 cm−1 for Be+−pH2 and Be+−oD2, respectively. The singly excited bending levels of the complexes with oH2 and pD2 appear in the 0+− block (673 and 512 cm−1 above the ground state for Be+−H2 and Be+−D2, respectively), whereas bending levels of Be+−pH2 and Be+−oD2 appear in the 1++ and 1−+ blocks as the components of asymmetry doublets slightly split by Coriolis interaction. The energies of one component are given in Table 2. Comparison with harmonic νb(h) frequencies (804 and 571 cm−1 for Be+−H2 and Be+−D2, respectively) indicates that the bending vibrational mode has large anharmonicity. Bending excitation increases the vibrationally averaged interfragment distance and decreases the diatomic bond length, but less than does the intermolecular stretching excitation. Variational results for all parity blocks with J = 0−4 for Be+− H2 and J = 0−2 for Be+−D2 were fitted using an A-reduced Watson Hamiltonian to determine effective spectroscopic constants for the nHH = 0 and nDD = 0 levels. Parameters are compiled in Table 3. As observed for other similar floppy

Table 2. Lowest Rovibrational Energy Levels of the Be+ Electrostatic Complexes with the Nuclear Spin-Isomers of H2 and D2a n

(ns,nb)

E, cm−1

⟨r⟩, Å

0 1 2 3 4

(0,0) (1,0) (2,0) (0,2) (3,0)

0

(0,1)

0 1 2 3 4

(0,0) (1,0) (2,0) (0,2) (3,0)

0

(0,1)

0 1 2 3 4

(0,0) (1,0) (2,0) (0,2) (3,0)

0

(0,1)

0 1 2 3 4

(0,0) (1,0) (2,0) (0,2) (3,0)

0

(0,1)

r,̅ Å

Be −pH2, J = 0 −2677.87 0.780 0.763 −2146.01 0.779 0.762 −1680.60 0.777 0.761 −1460.35 0.779 0.762 −1280.53 0.776 0.760 Be+−pH2, Jpipj = 1++ −1910.08 0.780 0.763 Be+−oH2, Jpipj = 1+− −2608.63 0.780 0.763 −2074.22 0.779 0.762 −1606.18 0.777 0.761 −1285.41 0.779 0.762 −1204.00 0.776 0.760 Be+−oH2, Jpipj = 0+− −2005.54 0.779 0.763 Be+−oD2, Jpipj = 0++ −2786.48 0.773 0.760 −2359.62 0.772 0.760 −1973.51 0.771 0.759 −1818.53 0.772 0.760 −1627.82 0.770 0.758 Be+−oD2, Jpipj = 1++ −2234.38 0.772 0.760 Be+−pD2, Jpipj = 1+− −2752.58 0.773 0.761 −2325.31 0.772 0.760 −1938.76 0.771 0.759 −1765.34 0.772 0.760 −1592.65 0.770 0.758 Be+−pD2, Jpipj= 0+− −2274.42 0.772 0.760 +

pipj

⟨R⟩, Å

R̅ , Å

1.844 1.950 2.073 1.956 2.219

1.828 1.903 1.991 1.926 2.095

1.869

1.888

1.843 1.949 2.073 1.935 2.218

1.828 1.903 1.990 1.909 2.094

1.893

1.872

1.826 1.906 1.995 1.898 2.096

1.815 1.871 1.934 1.879 2.006

1.857

1.843

1.826 1.906 1.995 1.894 2.096

1.815 1.871 1.934 1.876 2.006

1.858

1.844

++

Table 3. Predicted Spectroscopic Constants for Be+−H2 and Be+−D2 Obtained by Fitting the Calculated Energy Levels Derived from the PES Presented in the Current Work to a Watson A-Reduced Hamiltoniana A B C B̅ ΔJ × 104 ΔJK × 103

a

Be+−H2b

Be+−D2c

66.198(1) 3.0517(5) 2.8542(5) 2.952 95 3.0(1) −7.15(2)

32.1316(8) 1.8359(5) 1.7093(5) 1.7726(5) 1.0(2) −2.70(2)

Energies are given with respect to the ground dissociation limits with the H2/D2 molecule in its ground-state nHH = 0, j = 0.

Units for all parameters are cm−1. bFrom fits to Ka = 0 and 1, J = 0−4 levels. cFrom fits to Ka = 0 and 1, J = 0−2 levels.

same differences apply for the D−D bond in Be+−D2. The explanation relates to the fact that the vibrationally averaged intermolecular separation (⟨R⟩ and R̅ ) is appreciably larger than the equilibrium Re separation (e.g., by 0.068 and 0.052 Å for Be+−pH2, 0.050 and 0.039 Å for Be+−oD2). Therefore, when the separation is averaged over the zero-point excursions in the intermolecular modes, the Be+ ion perturbs the H2 molecule less than at the equilibrium separation, Re. For the same reason, intermolecular excitations that, on average, move the fragments further apart slightly decrease the vibrationally averaged H2 and D2 separations. Indeed, Table 2 shows that there is a clear inverse correlation between the vibrationally averaged r and R values. The lowest vibrational energy levels can be assigned according to the number of quanta in the decoupled intermolecular stretching (νs) and bending (νb) vibrational modes, guided by the nodal patterns of the rovibrational wave functions and the vibrationally averaged intermolecular separation (⟨R⟩ or R̅ ). Using the energies and assignments from Table 2, one can derive the fundamental stretching frequencies for Be+−pH2 and Be+−oD2 as 532 and 427 cm−1, respectively, and estimate their harmonic analogs as 598 and

atom−diatomic complexes, although the lower energy levels are adequately fit by the standard Watson Hamiltonian, the spectroscopic constants are influenced by the large-amplitude vibrational motions (particularly the intermolecular bend vibration), and the A constant in particular is difficult to relate to any sensible geometrical structure. Intramolecular Vibrational Frequency Shift. To estimate the shifts in the vibrational frequencies of H2 and D2 molecules in the Be+−H2 and Be+−D2 complexes, 2D variational calculations were performed. Energies and intensities were computed for transitions connecting the lowest rovibrational levels of the initial nHH = 0 and final nHH = 1 manifolds, as described elsewhere.22 Results for the most intense transitions (the lowest R branch transitions) from the ground levels of each complex (selection rules are J″ − J′ = 0, ± 1, p′i = −p″i , p′j = p″j ) are presented in Table 4. The energies of the initial E″ and final E′ levels are given with respect to the ground vibrationally adiabatic asymptotic limit, Be+ + H2 (nHH = 0,1, j = 0) or Be+ + D2 (nDD = 0,1, j = 0). It should be noted that there is a significant difference (82 cm−1, or 3% of De) between the nHH = 0 energies calculated within the 3D approach (cf. Table 2) and the 2D approximation. In the former, inclusion of the wave functions describing vibrational

a

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Table 4. Results for 2D Calculations of the Lower nHH/DD = 0 → nHH/DD = 1 R-Branch Transitions for the Be+ Electrostatic Complexes with the Nuclear Spin-Isomers of H2 and D2a complex

J″pi″pj″ → J″pi″pj″

Be −pH2 Be+−oH2 Be+−oD2 Be+−pD2

0 →1 1+− → 2−− 0++ → 1−+ 1+− → 2−−

+

++

−+

E″, cm−1

E′, cm−1

ΔνHH/DD, cm−1

−2596.05 −2526.65 −2703.18 −2669.22

−2913.03 −2840.95 −2928.22 −2891.76

−316.98 −314.30 −225.04 −222.54

molecule (Δq) calculated at equilibrium geometry using a natural bond orbital (NBO) analysis.58 Charge transfer data from ref 12 were obtained at the CCSD(T) level with the augcc-pVTZ basis set, whereas the present calculations used the same basis set with the MRCI+Q method. Comparison of experimental and theoretical data in Table 5 allows an indirect assessment of the present theoretical approach. It shows that variational calculations with the CCSD(T) PESs effectively reproduces dissociation energies (B+−H2 is an apparent exception, although the equilibrium measurements are likely related to the oH2 complex with D0 = 1298 cm−1; see ref 29) and interfragment distances R0 (though different definitions, ⟨R⟩ or R̅ , are used in Table 5). As well, the 2D variational approach tends to underestimate the H2 vibrational frequency shift by 4−8% (Table 5). Similar discrepancies might be expected for the vibrational shifts obtained here for the Be+−H2 and Be+−D2 complexes. The short range approach of the M+ ion to the H2 molecule in the T-shaped configuration results in Pauli repulsion between electrons in the hydrogen molecule’s σg bonding orbital and valence electrons on the M+ cation. For Be+−H2, the repulsion can be mitigated through hybridization of the singly occupied 2s orbital and the 2pz orbital, shifting the electron density away from the intermolecular region to the opposite side of the Be+ cation. This hybridization is accompanied by charge transfer from the σg bonding orbital to the hybrid orbital.12 The magnitude of Δq value provides a clue for assessing this effect (although the predicted magnitude of the charge transfer can be sensitive to the ab initio scheme). The large value of Δq for Be+−H2 (Table 5) indicates that charge transfer is substantial and that hybridization plays an important role in stabilizing the complex. A similar hybridization occurs for Mg+−H2,12 but the smaller s/pz energy gap for Be+ (3.96 eV) compared to that for Mg+ (4.22 eV) along with a shorter intermolecular bond makes the effect more important for Be+−H2 than Mg+−H2. Inspection of Table 5 shows that for the ions studied so far, Be+ forms the strongest bond with H2 and leads to the largest deformation of the attached hydrogen molecule. Variation of the structural, vibrational, and energetic parameters moving down the Periodic Table clearly reflect weakening of the intermolecular bond, due to increasing ionic radius (i.e., increasing exchange repulsion). However, the variations are different within each group. For example, the interfragment distance increases and the dissociation energy drops from Be+ to Mg+ much more than from Li+ to Na+, so already at the second row the alkali metal ion binds to H2 more strongly than does the alkailine-earth metal ion. Surprisingly, the perturbation of the H2 diatomic does not reflect the same trend: both the H−H bond elongation and vibrational frequency shift are larger

Initial E″ and final E′ energies are given with respect to the dissociation limits with H2/D2 molecule in its rotational ground-state nHH/DD = 0, j″ = 0 and nHH/DD = 1,j′ = 0, respectively.

a

excitations of the molecular fragment in the basis set effectively allows it to stretch, ensuring better sampling of the minimum region of the PES. In contrast, the 2D approximation keeps the diatomic subunit essentially unperturbed. According to the 2D method, the vibrational frequency shift originates from the difference in the effective two-dimensional PESs associated with distinct vibrational states of the diatomic molecule; i.e., it results from the influence of the diatomic vibrational excitation on the intermolecular interaction. The predicted band shifts, based on fitting the nHH = 0 and 1 2D levels to an A-reduced Watson Hamiltonian, are ΔνHH = −322.9 cm−1 for Be+−H2 and ΔνDD = −228.6 for Be+−D2. The difference in the origins for the Ka = 0−0 and Ka = 1−1 subbands, which are due to distinct nuclear spin-isomers, is predicted to be ΔA = 3.4 cm−1 for Be+−H2 and ΔA = 1.2 cm−1 for Be+−D2 and should easily be resolvable in their respective infrared photodissociation spectra. The shifts derived from the 2D analysis correspond closely with the results of the normalmode analysis, ΔνHH(h) = 320 cm−1 and ΔνDD(h) = 228 cm−1.

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DISCUSSION Information is now available for a broad range of electrostatic M+−H2 complexes with various metal and semimetal cations2,12 allowing one to draw correlations between their bonding properties and parameters relating to the H−H bond activation. Our present results complete the first two rows of the I, II, and III groups of the Periodic Table, allowing comparisons between complexes on (practically) the same footing. Table 5 summarizes the main parameters of the M+−H2 complexes containing the Li, Be, B, Na, Mg, and Al cations. Most data were taken from Table 2 in ref 12 and include the results of IR spectroscopic investigations and theoretical results obtained using the variational energy level calculations based on ab initio PESs (refs 19 and 21 for Li+−H2, ref 29 for B+−H2, ref 26 for Na+−H2, refs 27 and 25 for Al+−H2). Dissociation energies (D0) derived from clustering equilibrium measurements are also given.5,10,11 In addition to structural and spectroscopic parameters, Table 5 lists the partial charge transfer from the H2

Table 5. Structural and Energetic Parameters (Theory/Experiment) for Selected M+−H2 Electrostatic Complexes (Sources Given in Text) parameter −1

D0, cm R0, Å Δr, Å ΔνHH, cm−1 Δq, au Δq, au12

Li+−H2

Be+−H2

B+−H2

Na+−H2

Mg+−H2

Al+−H2

1732/− −/2.06 0.010/− −/108

2678 /− 1.83/− 0.027/− 323 /− 0.051

1254/1330 ± 70 2.28/2.26 0.014/− 204/222 0.031 0.022

842/860 ± 70 2.51/2.49 0.006/− 64/67

614/− 2.73/2.72 0.008/− 102/106 0.007 0.009

473/470 ± 50 3.07/3.04 0.005/− 62/66

0.006

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for Mg+−H2 than for Na+−H2. The variations for B+−H2 and Al+−H2 lie between the Li−Na and Be−Mg extremes. The present results should be useful for guiding future infrared photodissociation investigations. We predict that the bands associated with the Be+−H2 and Be+−D2 complexes should be red-shifted from the H2 and D2 fundamental vibrational transitions by 315 and 220 cm−1, respectively. Complexes with D2 may be particularly interesting because the dissociation energies for Be+−oD2 and Be+−pD2 (2786 and 2812 cm−1, respectively) are predicted to be close to the nDD = 0 → nDD = 1 transition energy (2769 cm−1). This means that, particularly for Be+−pD2, some of the lower rotational levels associated with upper nDD = 1 manifold may be bound rather than metastable and transitions to these levels may not appear in the photodissociation spectrum. Such threshold effects were observed for the Cr+−D2 complex and were used to deduce an accurate dissociation energy.13 Therefore, an experimental investigation of Be+−D2 would be extremely interesting for assessing the theoretical methodology and providing a benchmark binding energy. Even though the splitting between the transitions of the complexes formed by distinct nuclear spin-isomers of hydrogen and deuterium is much larger than the typical spectral resolution (0.02 cm−1), transitions associated with pH2 are usually difficult to detect. The reason is the quite large difference in stability of the M+−oH2 and M+−pH2 complexes (predicted to be ∼50 cm−1 for Be+−H2). As a result, the equilibrium of the ortho/para ligand exchange is shifted toward the oH2 complex, as explained in detail in ref 59. Indeed, the 40 cm−1 difference in stability of Mg+−oH2 and Mg+−pH2 made it impossible to discern the transitions of the latter complexes above the attainable signal-to-noise ratio.27 The analogous difference for Be+−oD2 and Be+−pD2 is only 25 cm−1, making it more favorable for detecting complexes containing both nuclear spin isomers, again in agreement with the Mg+ case. Finally, one should mention that it is impossible to obtain the A rotational constant (and an estimate of the H−H bond length) from analysis of the parallel νHH stretch band.2 Our present calculations should provide a reliable estimate for the deformation of H−H bond caused by the Be+ ion.

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■

4. Calculations show that Be+ is attached more strongly to H2 than neighboring cations (Li+ and B+) and that it affects the vibrational frequency and bond length of dihydrogen more profoundly than its neighbors. 5. The Be+−D2 system is predicted to exhibit a threshold in its infrared predissociation spectrum at a particular J level in the nDD = 1 manifold because the dissociation energy is slightly higher than the energy of the νDD stretch mode. Observation of this threshold would permit an accurate experimental determination of the dissociation energy.

AUTHOR INFORMATION

Corresponding Author

*A. A. Buchachenko. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by Russian Foundation for Basic Research under project No. 14-03-00422 and the Australian Research Council’s Discovery Project funding scheme (Project Number DP120100100). J. K. is grateful for the support from the U. S. National Science Foundation (Grant No. CHE1213322 to M. H. Alexander).

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REFERENCES

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SUMMARY AND CONCLUSIONS 1. A new RKHS interaction potential energy surface has been developed for the Be+−H2 electrostatic complex based on the CCSD(T) approach using the aug-ccpVQZ basis set extended with additional 3s3p2d2f1g bond functions placed halfway between H2 center of mass and Be cation. 2. The PES, combined with the accurate isotopic-invariant H2 potential energy curve, was used to calculate the energies and wave functions for the lower rovibrational states of Be+−H2 and Be+−D2. Effective dissociation energies for Be+−pH2 and Be+−oH2 are predicted to be 2678 and 2727 cm−1, respectively. Predicted dissociation energies for Be+−oD2 and Be+−pD2 are slightly larger due to reduced vibrational zero-point energy and are 2786 and 2812 cm−1, respectively. 3. On the basis of 2D rovibrational calculations, the νHH band of Be+−H2 is predicted to be shifted by −323 cm−1 with respect to the band of the free H2 molecule, whereas the corresponding shift for Be+−D2 is −229 cm−1. 6718



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